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On Generalized Bell Polynomials
Roberto B. Corcino,Cristina B. Corcino
Discrete Dynamics in Nature and Society , 2011, DOI: 10.1155/2011/623456
Abstract: It is shown that the sequence of the generalized Bell polynomials is convex under some restrictions of the parameters involved. A kind of recurrence relation for is established, and some numbers related to the generalized Bell numbers and their properties are investigated. 1. Introduction Hsu and Shiue [1] defined a kind of generalized Stirling number pair with three free parameters which is introduced via a pair of linear transformations between generalized factorials, viz, where (set of nonnegative integers), may be real or complex numbers with ( ) (0,？0,？0), and denotes the generalized factorial of the form In particular, with . Various well-known generalizations were obtained by special choices of the parameters and (cf. [1]), and the generalization of some properties of the classical Stirling numbers such as the recurrence relations the exponential generating function the explicit formula the congruence relation, and a kind of asymptotic expansion was established. As a follow-up study of these numbers, more properties were obtained in [2]. Furthermore, some combinatorial interpretations of were given in [3] in terms of occupancy distribution and drawing of balls from an urn. Hsu and Shiue [1] also defined a kind of generalized exponential polynomials in terms of generalized Stirling numbers with real or complex numbers as follows: We may call these polynomials generalized Bell polynomials. Note that when , we get the generalized Bell numbers. A kind of generating function of the sequence for the generalized exponential polynomials has been established by Hsu and Shiue, viz, where . In particular, (1.8) gives the generating function for the generalized Bell numbers: Note that, when , . Hence, If we define the polynomial as then its exponential generating function is given by We may call the -Bell polynomial. Hence, with , this yields the exponential generating function for the -Bell numbers. Now, if we use to denote the following limit: then, by (1.5), Also obtained by Hsu and Shiue is an explicit formula for of the form Consequently, with , we have Note that, by taking , (1.16) gives the explicit formula for -Bell polynomial. When , this gives a kind of the Dobinski formula for -Bell numbers. This reduces further to the Dobinski formula for -Bell numbers [4] when . Moreover, with , we get which is the Dobinski formula for the ordinary Bell numbers [5]. In this paper, a recurrence relation and convexity of the generalized Bell numbers will be established and some numbers related to will be investigated. Some theorems on -Bell polynomials will be
An Asymptotic Formula for -Bell Numbers with Real Arguments
Cristina B. Corcino,Roberto B. Corcino
ISRN Discrete Mathematics , 2013, DOI: 10.1155/2013/274697
Abstract:
An Asymptotic Formula for -Bell Numbers with Real Arguments
Cristina B. Corcino,Roberto B. Corcino
ISRN Discrete Mathematics , 2013, DOI: 10.1155/2013/274697
Abstract: The -Bell numbers are generalized using the concept of the Hankel contour. Some properties parallel to those of the ordinary Bell numbers are established. Moreover, an asymptotic approximation for -Bell numbers with real arguments is obtained. 1. Introduction The -Stirling numbers of the second kind, denoted by , are defined by Broder in [1], combinatorially, to be the number of partitions of the set into nonempty subsets, such that the numbers are in distinct subsets. Several properties of these numbers are established in [1–3]. Further generalization was established in [4] which is called -Stirling numbers. These numbers are equivalent to the -Whitney numbers of the second kind [5] and the Rucinski-Voigt numbers [6]. The sum of -Stirling numbers of the second kind for integral arguments was first considered by Corcino in [7] and was called the -Bell numbers. Corcino obtained an asymptotic approximation of -Bell numbers using the method of Moser and Wyman. Here, we use to denote the -Bell numbers; that is, In a followup study of Mez？ [8], the -Bell numbers were given more properties. One of these is the following exponential generating function: A more general form of Bell numbers, denoted by , was defined in [9] as where the parameters , , and are complex numbers with , , and . In this paper, we define the -Bell numbers with complex argument using the concept of Hankel contour and establish some properties parallel to those obtained by Mez？ in [8]. Moreover, an asymptotic formula of these numbers for real arguments will be derived using the method of Moser and Wyman [10]. 2. -Stirling Numbers of the Second Kind Graham et al. [11] proposed another way of generalizing the Stirling numbers by extending the range of values of the parameters and to complex numbers. This problem was first considered by Flajolet and Prodinger [12] by defining the classical Stirling numbers with complex arguments using the concept of Hankel contour. Recently, the -Stirling numbers with complex arguments, denoted by , were defined in [9] by means of the following integral representation over a Hankel contour : where and are complex numbers with , , and . We know that, for integral case, the -Stirling numbers of the second kind may be obtained by taking . Hence, using (4), we can define the second-kind -Stirling numbers with complex arguments as follows. Definition 1. The -Stirling numbers of the second kind of complex arguments and are defined by where is complex number with and , and the logarithm involved in the functions and is taken to be the principal branch. The Hankel
On r-Stirling Type Numbers of the First Kind
Cristina B. Corcino,Roberto B. Corcino
- , 2019, DOI: 10.12691/tjant-7-3-2
Abstract: Some combinatorial properties of -Stirling numbers are proved. Moreover, two asymptotic formulas for -Stirling numbers of the first kind derived using different methods are discussed and corresponding asymptotic formulas for the -Stirling type numbers of the first kind are obtained as corollaries
On Generalized Multi Poly-Euler and Multi Poly-Bernoulli Polynomials
Roberto B. Corcino,Hassan Jolany,Cristina B. Corcino,Takao Komatsu
Mathematics , 2015,
Abstract: In this paper, we establish more identities of generalized multi poly-Euler polynomials with three parameters and obtain a kind of symmetrized generalization of the polynomials. Moreover, generalized multi poly-Bernoulli polynomials are defined using multiple polylogarithm and derive some properties parallel to those of poly-Bernoulli polynomials. These are generalized further using the concept of Hurwitz-Lerch multiple zeta values.
More Properties on Multi Poly-Euler Polynomials
Hassan Jolany,Roberto B. Corcino
Mathematics , 2014, DOI: 10.1007/s40590-015-0061-y
Abstract: In this paper, we establish more properties of generalized poly-Euler polynomials with three parameters and we investigate a kind of symmetrized generalization of poly- Euler polynomials. Moreover, we introduce a more general form of multi poly-Euler polynomials and obtain some identities parallel to those of the generalized poly-Euler polynomials.
Explicit formula for generalization of Poly-Bernoulli numbers and polynomials with a,b,c parameters
Hassan Jolany,Roberto B. Corcino
Mathematics , 2011,
Abstract: In this paper we investigate special generalized Bernoulli polynomials with a,b,c parameters that generalize classical Bernoulli numbers and polynomials. The present paper deals with some recurrence formulae for the generalization of poly-Bernoulli numbers and polynomials with a,b,c parameters. Poly-Bernoulli numbers satisfy certain recurrence relationships which are used in many computations involving poly-Bernoulli numbers. Obtaining a closed formula for generalization of poly-Bernoulli numbers with a,b,c paramerers therefore seems to be a natural and important problem. By using the generalization of poly-Bernoulli polynomials with a,b,c parameters of negative index we define symmetrized generalization of poly-Bernoulli polynomials with a,b parameters of two variables and we prove duality property for them. Also by stirling numbers of the second kind we will find a closed formula for them. Furthermore we generalize the Arakawa-Kaneko Zeta functions and by using the Laplace-Mellin integral, we define generalization of Arakawa-Kaneko Zeta functions with a,b parameters and we obtain an interpolation formula for the generalization of poly-Bernoulli numbers and polynomials with a,b parameters. Furthermore we present a link between this type of Zeta functions and Dirichlet series. By our interpolation formula, we will interpolate the generalization of Arakawa-Kaneko Zeta functions with a,b parameters.
A -Analogue of Rucinski-Voigt Numbers
Roberto B. Corcino,Charles B. Montero
ISRN Discrete Mathematics , 2012, DOI: 10.5402/2012/592818
Abstract:
A -Analogue of Rucinski-Voigt Numbers
Roberto B. Corcino,Charles B. Montero
ISRN Discrete Mathematics , 2012, DOI: 10.5402/2012/592818
Abstract: A -analogue of Rucinski-Voigt numbers is defined by means of a recurrence relation, and some properties including the orthogonality and inverse relations with the -analogue of the limit of the differences of the generalized factorial are obtained. 1. Introduction Rucinski and Voigt [1] defined the numbers satisfying the relation where is the sequence and and proved that these numbers are asymptotically normal. We call these numbers Rucinski-Voigt numbers. Note that the classical Stirling numbers of the second kind in [2–4] and the -Stirling numbers of the second kind of？？Broder [5] can be expressed in terms of as follows: where and are the sequences and , respectively. With these observations, may be considered as certain generalization of the second kind Stirling-type numbers. Several properties of Rucinski-Voigt numbers can easily be established parallel to those in the classical Stirling numbers of the second kind. To mention a few, we have the triangular recurrence relation the exponential and rational generating function and explicit formulas The explicit formula in can be used to interpret as the number of ways to distribute distinct balls into the cells ( one ball at a time ), the first of which has distinct compartments and the last cell with distinct compartments, such that(i)the capacity of each compartment is unlimited;(ii)the first cells are nonempty. The other explicit formula can also be used to interpret as the number of ways of assigning people to groups of tables where all groups are occupied such that the first group contains distinct tables and the rest of the group each contains distinct tables. The Rucinski-Voigt numbers are nothing else but the -Whitney numbers of the second kind, denoted by , in Mez？ [6]. That is, . It is worth-mentioning that the -Whitney numbers of the second kind are generalization of Whitney numbers of the second kind in Benoumhani's papers [7–9]. On the other hand, the limit of the differences of the generalized factorial [10] was also known as a generalization of the Stirling numbers of the first kind. That is, all the first kind Stirling-type numbers may also be expressed in terms of by a special choice of the values of and . It was shown in [10] that where is the sequence . Recently, -analogue and -analogue of , denoted by and , respectively, were established by Corcino and Hererra in [10] and obtained several properties including the horizontal generating function for where The numbers are equivalent to the -Whitney numbers of the first kind, denoted by , in [6]. More precisely, . These numbers are
Generalized $q$-Stirling numbers and normal ordering
Roberto B. Corcino,Ken Joffaniel M. Gonzales,Richell O. Celeste
Mathematics , 2014,
Abstract: The normal ordering coefficients of strings consisting of $V,U$ which satisfy $UV=qVU+hV^s$ ($s\in\mathbb N$) are considered. These coefficients are studied in two contexts: first, as a multiple of a sequence satisfying a generalized recurrence, and second, as $q$-analogues of rook numbers under the row creation rule introduced by Goldman and Haglund. A number of properties are derived, including recurrences, expressions involving other $q$-analogues and explicit formulas. We also give a Dobinsky-type formula for the associated Bell numbers and the corresponding extension of Spivey's Bell number formula. The coefficients, viewed as rook numbers, are extended to the case $s\in\mathbb R$ via a modified rook model.
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